Stability of some difference equations with nonhyperbolic equilibria

Abstract

Our main goal in this study is to investigate the stability of some nonlinear difference equations of order two when the equilibrium of the equation is nonhyperbolic and linearized stability fails to apply.^ First we investigate the stability of the positive equilibrium of the system, known as May's Host Parasitoid Model:$$\left.\eqalign{x\sb{n+1} &= {{\alpha x\sb{n}}\over{1+\beta y\sb{n}}}\cr y\sb{n+1} &= {{\beta x\sb{n}y\sb{n}}\over{1+\beta y\sb{n}}}}\right\},n = 0,1,\...$$where $\alpha$ and $\beta$ are positive numbers and the initial conditions $x\sb0$ and $y\sb0$ are arbitrary positive numbers. We determine the stability nature of the positive equilibrium and we present a thorough analysis of the global behavior of the solutions of this system.^ Next we establish the stability nature of the equilibrium ${\cal E}$ of Lyness' equation$$z\sb{n+1} = {{z\sb{n}+\beta}\over{z\sb{n-1}}}, n= 0,1,\...$$where $\beta$ is a positive number and the initial conditions $z\sb{-1}$ and $z\sb0$ are arbitrary positive numbers. Note that the change of variables$$\left.\eqalign{z\sb{n} &= {\cal E}\cdot e\sp{x\sb{n}}\cr z\sb{n-1} &= {\cal E}\cdot e\sp{y\sb{n}}}\right\}\cr$$reduces Lyness' equation to an area preserving map and transforms ${\cal E}$ to the origin which is a nonhyperbolic equilibrium of the elliptic type. We then apply the Kolmogorov-Arnold-Moser theory to determine the stability character of ${\cal E}$.^ Finally we investigate the stability character of the equilibria and the boundedness and oscillatory behavior of the solutions of the difference equation$$w\sb{n+1}= {\max\{w\sbsp{n}{k}, A\}\over w\sb{n-1}}, n = 1,2,\...$$where$$A, k\in(0,\infty)$$and the initial conditions $w\sb0,\ w\sb1$ are arbitrary positive numbers. ^